Scalets, wavelets and (complex) turning point quantization
Carlos R. Handy, Harold A. Brooks · Journal of Physics A Mathematical and General · 2001
Despite the many successes of wavelet analysis in image and signal processing, the incorporation of continuous wavelet transform theory within quantum mechanics has lacked a compelling, first principles , motivating analytical framework, until now. For arbitrary one-dimensional rational fraction Hamiltonians, we develop a simple, unified formalism, which clearly underscores the complementary, and mutually interdependent, role played by moment quantization theory (i.e. via scalets, as defined herein) and wavelets. This analysis involves no approximation of the Hamiltonian within the (equivalent) wavelet space, and emphasizes the importance of (complex) multiple turning point contributions in the quantization process. We apply the method to three illustrative examples. These include the (double-well) quartic anharmonic oscillator potential problem, V ( x ) = Z 2 x 2 + gx 4 , the quartic potential, V ( x ) = x 4 , and the very interesting and significant non-Hermitian potential V ( x ) = -(i x ) 3 , recently studied by Bender and Boettcher.